MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem30 Structured version   Visualization version   GIF version

Theorem fin23lem30 9753
Description: Lemma for fin23 9800. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem30 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fin23lem.e . 2 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4510 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
32biimpi 217 . 2 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
4 simpr 485 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5 fin23lem.d . . . . . . . . . . . 12 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
65funmpt2 6391 . . . . . . . . . . 11 Fun 𝑅
7 funco 6392 . . . . . . . . . . 11 ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡𝑅))
84, 6, 7sylancl 586 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡𝑅))
9 elunirn 7004 . . . . . . . . . 10 (Fun (𝑡𝑅) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
108, 9syl 17 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
11 dmcoss 5841 . . . . . . . . . . . 12 dom (𝑡𝑅) ⊆ dom 𝑅
1211sseli 3967 . . . . . . . . . . 11 (𝑏 ∈ dom (𝑡𝑅) → 𝑏 ∈ dom 𝑅)
13 fvco 6756 . . . . . . . . . . . . . . . 16 ((Fun 𝑅𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
146, 13mpan 686 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom 𝑅 → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1514adantl 482 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1615eleq2d 2903 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅𝑏))))
17 incom 4182 . . . . . . . . . . . . . . . 16 ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏)))
18 difss 4112 . . . . . . . . . . . . . . . . . . . . . . 23 (ω ∖ 𝑃) ⊆ ω
19 ominf 8719 . . . . . . . . . . . . . . . . . . . . . . . . 25 ¬ ω ∈ Fin
20 fin23lem.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
2120ssrab3 4061 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑃 ⊆ ω
22 undif 4433 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
2321, 22mpbi 231 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃 ∪ (ω ∖ 𝑃)) = ω
24 unfi 8774 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
2523, 24eqeltrrid 2923 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
2625ex 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
2719, 26mtoi 200 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
2827ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
295fin23lem22 9738 . . . . . . . . . . . . . . . . . . . . . . 23 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
3018, 28, 29sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
31 f1of 6612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
3230, 31syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
33 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
3432fdmd 6520 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
3533, 34eleqtrd 2920 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
3632, 35ffvelrnd 6848 . . . . . . . . . . . . . . . . . . . 20 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ (ω ∖ 𝑃))
3736eldifbd 3953 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ 𝑃)
3820eleq2i 2909 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ 𝑃 ↔ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
3937, 38sylnib 329 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
4036eldifad 3952 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ ω)
41 fveq2 6667 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑅𝑏) → (𝑡𝑣) = (𝑡‘(𝑅𝑏)))
4241sseq2d 4003 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝑅𝑏) → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4342elrab3 3685 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ ω → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4440, 43syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4539, 44mtbid 325 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)))
46 fin23lem.a . . . . . . . . . . . . . . . . . . 19 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
4746fin23lem20 9748 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑏) ∈ ω → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
4840, 47syl 17 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
49 orel1 884 . . . . . . . . . . . . . . . . 17 ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) → (( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
5045, 48, 49sylc 65 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅)
5117, 50syl5eq 2873 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅)
52 disj 4402 . . . . . . . . . . . . . . 15 (((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
5351, 52sylib 219 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
54 rsp 3210 . . . . . . . . . . . . . 14 (∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5553, 54syl 17 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5616, 55sylbid 241 . . . . . . . . . . . 12 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
5756ex 413 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
5812, 57syl5 34 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
5958rexlimdv 3288 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
6010, 59sylbid 241 . . . . . . . 8 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) → ¬ 𝑎 ran 𝑈))
6160ralrimiv 3186 . . . . . . 7 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
62 disj 4402 . . . . . . 7 (( ran (𝑡𝑅) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
6361, 62sylibr 235 . . . . . 6 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅)
64 rneq 5805 . . . . . . . . 9 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6564unieqd 4847 . . . . . . . 8 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6665ineq1d 4192 . . . . . . 7 (𝑍 = (𝑡𝑅) → ( ran 𝑍 ran 𝑈) = ( ran (𝑡𝑅) ∩ ran 𝑈))
6766eqeq1d 2828 . . . . . 6 (𝑍 = (𝑡𝑅) → (( ran 𝑍 ran 𝑈) = ∅ ↔ ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅))
6863, 67syl5ibr 247 . . . . 5 (𝑍 = (𝑡𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran 𝑍 ran 𝑈) = ∅))
6968expd 416 . . . 4 (𝑍 = (𝑡𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)))
7069impcom 408 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
71 rneq 5805 . . . . . . . 8 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7271unieqd 4847 . . . . . . 7 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7372ineq1d 4192 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈))
74 rncoss 5842 . . . . . . . 8 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
7574unissi 4860 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
76 disj 4402 . . . . . . . 8 (( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ¬ 𝑎 ran 𝑈)
77 eluniab 4848 . . . . . . . . . 10 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)))
78 eleq2 2906 . . . . . . . . . . . . . 14 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈)))
79 eldifn 4108 . . . . . . . . . . . . . 14 (𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈) → ¬ 𝑎 ran 𝑈)
8078, 79syl6bi 254 . . . . . . . . . . . . 13 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8180rexlimivw 3287 . . . . . . . . . . . 12 (∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8281impcom 408 . . . . . . . . . . 11 ((𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8382exlimiv 1924 . . . . . . . . . 10 (∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8477, 83sylbi 218 . . . . . . . . 9 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} → ¬ 𝑎 ran 𝑈)
85 eqid 2826 . . . . . . . . . . 11 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
8685rnmpt 5826 . . . . . . . . . 10 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8786unieqi 4846 . . . . . . . . 9 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8884, 87eleq2s 2936 . . . . . . . 8 (𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8976, 88mprgbir 3158 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅
90 ssdisj 4412 . . . . . . 7 (( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∧ ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅) → ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅)
9175, 89, 90mp2an 688 . . . . . 6 ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅
9273, 91syl6eq 2877 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ∅)
9392a1d 25 . . . 4 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9493adantl 482 . . 3 ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9570, 94jaoi 853 . 2 (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
961, 3, 95mp2b 10 1 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wral 3143  wrex 3144  {crab 3147  Vcvv 3500  cdif 3937  cun 3938  cin 3939  wss 3940  c0 4295  ifcif 4470  𝒫 cpw 4542   cuni 4837   cint 4874   class class class wbr 5063  cmpt 5143  dom cdm 5554  ran crn 5555  ccom 5558  suc csuc 6191  Fun wfun 6346  wf 6348  1-1-ontowf1o 6351  cfv 6352  crio 7105  (class class class)co 7148  cmpo 7150  ωcom 7568  seqωcseqom 8074  m cmap 8396  cen 8495  Fincfn 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-seqom 8075  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357
This theorem is referenced by:  fin23lem31  9754
  Copyright terms: Public domain W3C validator